Sufficient Conditions of Polynomial Solvability of the Two-Machine Preemptive Routing Open Shop on a Tree
The routing open shop problem with preemption allowed is a natural combination of the metric TSP problem and the classical preemptive open shop scheduling problem. While metric TSP is strongly NP-hard, the preemptive open shop is polynomially solvable for any (even unbounded) number of machines. The previous research on the preemptive routing open shop is mostly focused on the case with just two nodes of the transportation network (problem on a link). It is known to be strongly NP-hard in the case of an unbounded number of machines and polynomially solvable for the two-machine case. The algorithmic complexity of both two-machine problem on a triangular network and a three-machine problem with two nodes are still unknown. The problem with a general transportation network is a generalization of the metric TSP and therefore is strongly NP-hard.
We describe a wide polynomially solvable subclass of the preemptive routing open shop on a tree. This class allows building an optimal schedule with at most one preemption in linear time. For any instance from that class optimal makespan coincides with the standard lower bound. Therefore, the result, previously known for the problem on a link, is generalized on a special case on an arbitrary tree. The algorithmic complexity of the general case of the two-machine problem on a tree remains unknown.
KeywordsScheduling Routing open shop Preemption Polynomially solvable subclass Overloaded node Overloaded edge
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